5.8 Graph Matching Example: set of worker assign to a set of task Four tasks are to be assigned to four tasks workers. workers A Worker I is qualified to do tasks b and o B Worker 2 is qualified to do tasks AC and D Worker 3 is qualified to do tasks B and d 4 Worker 4 is qualified to do task a and c Can all 4 workers be assigned to different tasks for which they are qualified?
5.8 Graph Matching ▪ Example: Set of worker assign to a set of task ▪ Four tasks are to be assigned to four workers. ▪ – Worker 1 is qualified to do tasks B and C ▪ – Worker 2 is qualified to do tasks A,C and D ▪ – Worker 3 is qualified to do tasks B and D ▪ – Worker 4 is qualified to do task A and C. ▪ Can all 4 workers be assigned to different tasks for which they are qualified?
Example 2: The Marriage Problem: Given a set of men, each of whom knows some women from a given set of women, under what conditions is it possible for all men to marry women they know? Four men each know some of four women Peter knows Mary and Ann Kevin knows Mary, Ann, Rose and Tina Brian knows Mary and Ann Fred knows Ann Is it possible for all the men to marry women they know? Graph matching ary eter ose Brian ●Tina Fred
▪ Example 2: The Marriage Problem:Given a set of men, each of whom knows some women from a given set of women, under what conditions is it possible for all men to marry women they know? ▪ Four men each know some of four women ▪ Peter knows Mary and Ann ▪ Kevin knows Mary, Ann, Rose and Tina ▪ Brian knows Mary and Ann ▪ Fred knows Ann ▪ Is it possible for all the men to marry women they know? ▪ Graph Matching
Definition 36: A matching M in a graph g(v; e)is a subset of the edge set e such that no two edges in M are incident on the same vertex. The size of a matching M is the number of edges in M. For a graph G(v; E), a matching of maximum size is called a maximum matching M1={el,e7},M2={el,e2,e5},M3 Rel, e2, e5, e6and M4 e2 Rel, e2, e7, e8 are matching. M3 and m4 are maximum matching 98
▪ Definition 36: A matching M in a graph G(V;E) is a subset of the edge set E such that no two edges in M are incident on the same vertex. The size of a matching M is the number of edges in M. For a graph G(V;E), a matching of maximum size is called a maximum matching. M1={e1,e7},M2={e1,e2,e5},M3 ={e1,e2,e5,e6}and M4= {e1,e2,e7,e8} are matching. M3 and M4 are maximum matching
Definition 37: If M is a matching in a graph G. a vertex v is said to be m-saturated if there is an edge in M incident on v. Vertex v is said to be m-unsaturated if there is no edge in m incident onv M={el,e7},M3={el,e2,e5,e6} MI-saturated: v /e2 MI-unsaturated: u M3-saturated: uv e
▪ Definition 37: If M is a matching in a graph G, a vertex v is said to be M-saturated if there is an edge in M incident on v. Vertex v is said to be M-unsaturated if there is no edge in M incident on v. M1={e1,e7}, M3={e1,e2,e5,e6} M1-saturated : v M1-unsaturated: u M3-saturated:u,v
2 M={el,e3,e7,e8}, MIel, e3, e5, e61 e 3 MI and m are perfect matching. e e e8 e 62 Definition 38: A matching M of g is perfect if all vertices of g are M-saturated. If g(vi;v2)is a 8 bipartite graph then a matching M of g that saturates all the vertices in VI is called a complete matching from VI to V2
▪ Definition 38: A matching M of G is perfect if all vertices of G are M-saturated. If G(V1;V2) is a bipartite graph then a matching M of G that saturates all the vertices in V1 is called a complete matching from V1 to V2. M={e1,e3,e7,e8}, M1={e1,e3,e5,e6} M1 and M are perfect matching
u3 U2 M={{vl,ul},{v2,u2},{v3,u3} M is a complete matching from Vi to v2, but it is not a complete matching from v2 to V1
▪ M={{v1,u1},{v2,u2},{v3,u3}} ▪ M is a complete matching from V1 to V2, but it is not a complete matching from V2 to V1
Definition 39: Given a matching M in a graph G, a M-alternating path(cycle) is a path(cycle) in g whose edges are alternately in m and outside ofm(i.e. if an edge of the path is in M, the next edge is outside M and vice versa). A M alternating path whose end vertices are M unsaturated is called an M-augmenting path. 2 vO y y
▪ Definition 39: Given a matching M in a graph G, a M-alternating path (cycle) is a path (cycle) in G whose edges are alternately in M and outside of M (i.e. if an edge of the path is in M, the next edge is outside M and vice versa). A Malternating path whose end vertices are Munsaturated is called an M-augmenting path
Theorem 5.25: Mis a maximal matching of G iff there is no augmenting path relative to M. Proof:(1) There is no augmenting path relative to M, we prove m is a maximal matching ofG Suppose m andn are matching with MN To see that men contains an augmenting path relative to M we consider G=(V, MEN) l≤dc;(v)≤2 Since mn, men has more edges from N than M and hence has at least one augmenting path relative to M contradiction
▪ Theorem 5.25：M is a maximal matching of G iff there is no augmenting path relative to M. ▪ Proof: (1) There is no augmenting path relative to M, we prove M is a maximal matching of G . ▪ Suppose M and N are matching with |M|<|N|. ▪ To see that MN contains an augmenting path relative to M we consider G’ = (V’, MN ) ▪ 1≤dG’(v)≤2 ▪ Since |M|<|N|, MN has more edges from N than M and hence has at least one augmenting path relative to M ▪ contradiction
(2)If M is a maximal matching of G then there is no augmenting path relative to M Assume that there exists an M-augmenting path To see that Mep is a matching of g and MOpIMI IMOp is a matching of g Vel, e2EMOp, el and e2 are not adjacent 2)M⊕p|>M MOpMMU p)- Mnp=MU pl-Mnpl M+p-2 Mnp) =|MH+1
▪ (2)If M is a maximal matching of G then there is no augmenting path relative to M ▪ Assume that there exists an M-augmenting path p. ▪ To see that Mp is a matching of G and |Mp|>|M| ▪ 1)Mp is a matching of G ▪ e1,e2Mp, e1 and e2 are not adjacent. ▪ 2) |Mp|>|M| ▪ |Mp|=|(M∪p)-(M∩p)|= |(M∪p)|-|(M∩p)| ▪ =|M|+|p|-2|(M∩p)| ▪ =|M|+1
graph ices s cv abset of to some U 3 u2 U2 1VV)-V∈v|u∈,uv∈EU丿 A={v1,v3},N(A)={v2,v6,v4} A1={vl,v4},N(A1)={v2,v6,v4,v3, V5, V1) vs
▪ Definition 40: Given a bipartite graph G(V1;V2), and a subset of vertices S V, the neighborhood N(S) is the subset of vertices of V that are adjacent to some vertex in S, i.e. ▪ N(S) ={vV|uS,{u,v}E(G)} A={v1,v3},N(A)={v2,v6,v4} A1={v1,v4},N(A1 )={v2,v6,v4,v3, v5,v1}